The Fano variety of lines on singular cyclic cubic fourfolds

We study the symplectic resolution of the Fano variety of lines on some singular cyclic cubic fourfolds, i.e. cubic fourfolds arising as cyclic 3:1 cover of ${\mathbb{P}}^4$ branched along a cubic threefold. In particular we are interested in the geometry of these varieties in the case of cyclic cubic fourfolds branched along a cubic threefold having one isolated singularity of type $A_i$ for $i=2,3,4$. On these symplectic resolutions we find a non-symplectic automorphism of order three induced by the covering automorphism.